The declaration weight_polynomial_decay_summable asserts that polynomial decay of prime weights with exponent at least 1 implies square-summability of those weights.
(1) In plain English: given a function lamP assigning a real weight to each prime and a real ε ≥ 0 such that |lamP p| ≤ C / p^(1+ε) for some C > 0 and all primes p, the series ∑_p (lamP p)^2 converges.
(2) In Recognition Science this supplies a concrete sufficient condition for the bandwidth-derived decay WeightSquareSummable needed to place the cost operator T_J on a legitimate spectral footing inside the Hilbert space of finite-support states.
(3) The formal statement is a theorem with implicit parameters lamP : Nat.Primes → ℝ and ε : ℝ, explicit hypotheses hε : 0 ≤ ε and h : WeightDecayPolynomial lamP ε, and conclusion WeightSquareSummable lamP. The proof proceeds by comparison: the bound produces a pointwise majorant C² / p^(2(1+ε)) whose summability over primes follows from the injective pull-back of Real.summable_one_div_nat_rpow and scaling.
(4) Visible dependencies and certificates inside the supplied source are the definitions WeightDecayPolynomial and WeightSquareSummable, the implication chain regularity_chain, and the master certificate cost_operator_regularity_certificate that records the theorem as one of its three structural facts.
(5) The declaration does not prove essential self-adjointness, compact resolvent, or trace-class membership of the heat kernel; those remain open sub-conjectures encoded as hypothesis structures EssentialSelfAdjointness, CompactResolvent and TraceClassHeatKernel.